No abstract

No abstract

No abstract

We consider a kernel-type nonparametric estimator of the intensity function of a cyclic Poisson process when the period is unknown. We assume that only a single realization of the Poisson process is observed in a bounded window which expands in time. We compute the asymptotic bias, variance, and the mean squared error of the estimator when the window indefinitely expands.

Abstract-We investigate the second order accuracy of the M out of N bootstrap for a Studentized trimmed mean using the Edgeworth expansion derived in a previous paper. Some simulations, which support our theoretical results, are also given. The effect of extrapolation in conjunction with the M out of N bootstrap for Studentized trimmed means is briefly discussed. As an auxiliary result we obtain a Bahadur's type representation for an M out of N bootstrap quantile. 1. INTRODUCTION This article is closely connected with our previous paper [11], where the validity of the one-term Edgeworth expansion (EE) and the empirical Edgeworth expansion (EEE) for a Studentized trimmed mean was established and explicit formulas for the leading terms of the EE were obtained. We study two second-order approximations to the distribution function (df ) of a Studentized trimmed mean: EE and bootstrap.During the past twenty five years the attention of many authors was focussed on Efron's bootstrap ([8]), and nowadays there exists a voluminous literature on this topic. The consistency of the standard nonparametric, or naive, bootstrap was proved for many interesting statistics, at least for the asymptotically normal ones (see [1,4,6,7,12,13,15,25] and references therein). One of the main reasons of interest in the bootstrap and its application in statistics is the second order accuracy property: under proper conditions the bootstrap approximation to the distribution function (df ) of a pivotal statistic is more accurate than the normal one. This beneficial property of the bootstrap was proved for the sample mean [25], for the class of Hoeffding's U -statistics (cf. [15]) and for some other statistics. The usual way to prove this fact is based on the congruence of the one-term EE for the pivotal statistic (in the 'real world') and the EE for its bootstrap counterpart (in the 'bootstrap world'). Typically the structure of the one-term EE for the bootstrapped statistic is the same as the one for the pivotal statistic, when the parameters of the formula of the first leading term of EE are replaced by their empirical counterparts (plug-in estimators). So, the application of a relevant version of the Law of Large Numbers implies the second order accuracy of the bootstrap (cf. [12,15,25]). However, the case of the trimmed mean is a special one. The problem is connected with the difficulty in obtaining the explicit formula for the oneterm EE (cf. [13]).In this paper we establish the validity of a one-term EE for the bootstrapped Studentized trimmed mean (in the bootstrap world). We also obtain an explicit formula for the M −1/2 -term (correcting for